The ending isn't really a surprise. You must have figured out that the book would end with Douglass getting his freedom, right? Plus, even if you didn't, Garrison's preface gives it away. So it's not like the "getting free" part is the big climax.
Instead, the ending challenges us to think about what "freedom" really means. Douglass wants us to understand that slavery is wrong and should be abolished, of course, but he also wants to show that freedom is something more than a legal category. For one thing, even when Douglass does get his freedom and goes to New York, he's lost and alone. Without the help of his friends, he wouldn't have been able to make a new start.
Even more than that, though, Douglass's real triumph is in finding his voice, both as a public speaker and as an author. We think that's why the book ends by returning to the moment that Garrison talks about in the preface, when Douglass first stands up and begins telling the crowd about his experiences.
In a way, the real happy ending might just be the fact that the book you're reading exists at all. After all, Douglass tells us that at first he didn't want to stand up in front of a crowd of assembled white people because he still, on the inside, felt like a slave. And just as giving that speech shows that he's finally become free on the inside, the fact that he follows it up by writing this book makes the same point: after freeing his body, becoming an author shows that he has freed his mind. Yet this also suggests that getting his freedom hasn't really been the end of his journey.
In fact, getting his freedom is almost anti-climactic, since slavery continues. Getting his freedom is in a way the beginning of Douglass's story, the beginning of his life as an activist working to end slavery and the beginning of the long road to freedom that African-Americans have traveled ever since.